5. A car rental agency has two rental locations. Cars picked up at one location can
be returned at either location. After a some time, the agency observed that per
week about 10% of the clients renting a car at location #1 return the rental car
to location #2, and that 15% of the clients renting at location #2 return the car
to location #1.
(a) Summarize this information in a 2-state Markov chain $\vec{v}_{n+1} = A\vec{v}_n$. Be sure
to identify the states of the system and to write down the transition matrix A,
and to explain what the entries of $\vec{v}_n$ mean.
(b) Verify that A is power convergent and find $\lim_{n\to\infty} A^n$.
(c) Assuming that the above trend continues, what is the distribution of the cars
at the rental locations in the long run?
